The effect of block-wise feedback on the throughput-delay
trade-off in streaming
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Joshi, Gauri, Yuval Kochman, and Gregory W. Wornell. “The
Effect of Block-Wise Feedback on the Throughput-Delay TradeOff in Streaming.” 2014 IEEE Conference on Computer
Communications Workshops (INFOCOM WKSHPS) (April 2014).
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http://dx.doi.org/10.1109/INFCOMW.2014.6849236
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arXiv:1403.0259v1 [cs.IT] 2 Mar 2014
The Effect of Block-wise Feedback on the
Throughput-Delay Trade-off in Streaming
Gauri Joshi
Yuval Kochman
Gregory W. Wornell
EECS Dept., MIT
Cambridge, MA 02139, USA
Email: gauri@mit.edu
School of CSE, HUJI
Jerusalem, Israel
Email: yuvalko@cs.huji.ac.il
EECS Dept., MIT
Cambridge, MA 02139, USA
Email: gww@mit.edu
Abstract—Unlike traditional file transfer where only total delay
matters, streaming applications impose delay constraints on each
packet and require them to be in order. To achieve fast in-order
packet decoding, we have to compromise on the throughput. We
study this trade-off between throughput and in-order decoding
delay, and in particular how it is affected by the frequency of
block-wise feedback to the source. When there is immediate
feedback, we can achieve the optimal throughput and delay
simultaneously. But as the feedback delay increases, we have
to compromise on at least one of these metrics. We present a
spectrum of coding schemes that span different points on the
throughput-delay trade-off. Depending upon the delay-sensitivity
and bandwidth limitations of the application, one can choose an
appropriate operating point on this trade-off.
I. I NTRODUCTION
A. Motivation
A recent report [1] shows that 62% of the Internet traffic in
North America comes from real-time streaming applications
such as NetFlix (28.88%) and YouTube (15.43%). Streaming
traffic consumes such a large fraction of Internet bandwidth
because video files inherently have a larger size than other
forms of data. Thus, there is a need to develop transmission
schemes which can ensure a high quality of experience to the
user, with efficient use of available bandwidth.
Unlike traditional file transfer where only total delay matters, streaming imposes delay constraints on each individual
packet. Further, many applications require in-order playback
of packets at the receiver. Packets received out of order
are buffered until the missing packets in the sequence are
successfully decoded. In audio and video applications some
packets can be dropped without affecting the streaming quality.
However, other applications such as remote desktop, and
collaborative tools such as Dropbox and Google Docs have
strict order constraints on packets, where packets represent
instructions that need to be executed in order at the receiver.
To ensure that packets are decoded in order, the transmission
scheme must give higher priority to older packets that were
delayed, or received in error due to channel noise. However,
repeating old packets instead of transmitting new packets
results in a loss in the overall rate at which packets are
delivered to the user, that is, the throughput. Thus there
is a fundamental trade-off between throughput and in-order
decoding delay.
The throughput loss incurred to achieve low in-order decoding delay can be significantly reduced if the source receives
feedback about packet losses, and thus can adapt its future
transmission strategy to strike the right balance between old
and new packets. We study this interplay between feedback
and the throughput-delay trade-off.
B. Previous Work
Only a few papers in literature have analyzed streaming codes. Fountain codes [2] are capacity-achieving erasure
codes, but they are not suitable for streaming because the
decoding delay is proportional to the size of the data. Streaming codes without feedback for constrained channels such
as adversarial and cyclic burst erasure channels were first
proposed in [3], and also extensive explored in [4], [5]. The
thesis [3] also proposed codes for more general erasure models
and analyzed their decoding delay. Decoding delay has also
been analyzed studied in [6], [7] in a multicast scenario with
immediate feedback to the source.
However, decoding delay does not capture in order packet
delivery which is required for streaming applications. This
aspect is captured in the delay metrics in [8] and [9], which
consider that packets are played in-order at the receiver.
The authors in [8] analyze the throughput-delay trade-off for
uncoded packet transmission over a channel with long feedback delay. In [9] we propose coding schemes that minimize
playback delay in point-to-point streaming for the no feedback
and immediate feedback cases. However, the case of blockwise feedback to the source remains to be explored.
C. Our Contributions
In this paper we consider this unexplored problem of how
to effectively utilize block-wise feedback to the source to
ensure in-order packet delivery to the user. In contrast to
playback delay considered in [8] and [9], we propose a more
versatile delay metric called the in-order decoding exponent.
This metric captures the burstiness in the in-order decoding of
packet for applications which require packets in-order, but do
not necessarily play them at a constant rate.
When there is immediate feedback, we can achieve the best
throughput-delay trade-off. But when the feedback comes in
blocks, we have to compromise on the throughput to ensure
fast in-order decoding. We present a spectrum of coding
2
schemes that span different points on the throughput-delay
trade-off. Depending upon the delay-sensitivity, and bandwidth
limitations of the application, one can choose an appropriate
operating point on this trade-off. The proposed codes can be
shown to be optimal over a broad class of schemes for the no
feedback, and small feedback delay cases.
II. P ROBLEM S ETUP
A. System Model
We consider a point-to-point packet streaming scenario
where the source has a large stream of packets s1 , s2 , · · · , sn .
The encoder creates a coded packet yn = f (s1 , s2 ..sn ) in
each slot n and transmits it over the channel. The encoding
function f is known to the receiver. For example, if yn is
a linear combination of the source packets, the coefficients
are included in the transmitted packet so that the receiver
can use them to decode the source packets from the coded
combination. Without loss of generality, we can assume that
yn is a linear combination of the source packets.
We consider an i.i.d. packet erasure channel where every
transmitted packet is correctly received with probability p, and
otherwise received in error and discarded. An erasure channel
is a good model when encoded packets have a set of checksum
bits that can be used to verify with high probability whether
the received packet is error-free.
The receiver application requires the stream of packets to be
in order. Packets received out of order are buffered until the
missing packets in the sequence are decoded. Due to this inorder property, the transmitter can stop including sk in coded
packets when it knows that the receiver can decode sk once
all si for i < k are decoded. We refer such packets as “seen”
packets. The notion of “seen” is defined formally as follows.
Definition 1 (Seen Packets). A packet sk is said to be “seen”
by the transmitter when it knows that a coded combination
that only includes sk and packets si for 1 ≤ i ≤ k is received
successfully.
We consider that the source receives block-wise feedback
about channel erasures after every d slots. Thus, before transmitting in slot kd + 1, for all integers k ≥ 1, the source
knows about the erasures in slots (k − 1)d + 1 to kd. It can
use this information to adapt its transmission strategy in slot
kd + 1. Block-wise feedback can be used to model a halfduplex communication channel where after every d slots of
packet transmission, the channel is reserved for the receiver
to send feedback about the status of decoding. Note that the
feedback can be used to estimate p, the probability of success
of the erasure channel, when it is unknown to the source.
B. Throughput and Delay Metrics
We consider two metrics to measure the quality of streaming, the throughput τ and in-order decoding exponent λ. The
throughput is the rate at which “innovative” coded packets are
received. A coded packet is said to be “innovative” if it is
linear independent with respect to the coded packets received
until then. The bandwidth required is proportional to 1/τ .
The throughput captures the overall rate at which packets go
through the channel, irrespective of the order. The in-order
decoding aspect is captured by a metric called the in-order
decoding exponent λ which is defined as follows.
Definition 2 (In-order Decoding Exponent). Let T be the
time between two successive instants of decoding one or more
packets in-order. Then the in-order decoding exponent λ is
log Pr(T > n)
.
(1)
n→∞
n
.
The relation (1) can also be stated as Pr(T > n) = e−nλ
.
where = stands for asymptotic equality defined in [10, Page
63]. The in-order decoding exponent captures the burstiness
in packet decoding. For example, if the streaming application
plays one in-order packet in every slot, and if there are
b packets in the receiver buffer, then the probability of an
interruption in playback is proportional to e−λb .
In this paper we analyze how the trade-off between τ and
λ is affected by the block-wise feedback delay d. We first
consider the extreme cases of immediate feedback (d = 1)
and no feedback (d = ∞) in Section III and Section IV
respectively. This gives us insights into the analysis of the
(τ, λ) trade-off for general d in Section V.
λ , − lim
III. I MMEDIATE F EEDBACK
In the immediate feedback (d = 1) case, the source has
complete knowledge of past erasures before transmitting each
packet. We can show that a simple automatic-repeat-request
(ARQ) scheme is optimal in both τ and λ. In this scheme, the
source transmits the lowest index unseen packet, and repeats
it until the packet successfully goes through the channel.
Since a new packet is received in every successful slot,
the throughput τ = p, the success probability of the erasure
channel. The ARQ scheme is throughput-optimal because the
throughput τ = p is equal to the information-theoretic capacity
of the erasure channel [10]. Moreover, it also gives the optimal
the in-order decoding exponent λ because one in-order packet
is decoded in every successful slot. To find λ, first observe
that the tail distribution of the time T , the interval between
successive in-order decoding instants is,
Pr(T > n) = (1 − p)n
(2)
Substituting this in Definition 2 we get the exponent λ =
− log(1−p). Based on this analysis of the immediate feedback
case, we can find limits on the range of achievable (τ, λ) for
any feedback delay d as follows.
Lemma 1. The throughput and delay metrics (τ, λ) achievable
for any feedback delay d lie in the region 0 ≤ τ ≤ ρ, and
0 ≤ λ ≤ − log(1 − p).
Proof: When feedback is received after blocks of d > 1
slots, the source has less knowledge about past erasures than
in the immediate feedback (d = 1) case. Thus, the (τ, λ) tradeoff when d > 1 is always worse than (τ, λ) = (p, − log(1−p))
the optimal trade-off for the immediate feedback (d = 1) case.
3
IV. N O F EEDBACK
Now we consider the other extreme case (d = ∞), where
there is no feedback to the source about channel erasures. We
propose a coding scheme and prove that it gives the best (τ, λ)
trade-off among a class of codes called full-rank codes which
are defined as follows.
Definition 3 (Full-rank Codes). In slot n we transmit a linear
combination of all packets s1 to sV [n] , where the coefficients
are chosen from a large enough field such that the coded
combinations are independent with high probability. We refer
to V [n] as the transmit index in slot n.
Conjecture 1. Since the packets are required in-order at the
receiver, we believe that given transmit index V [n], there is
no loss of generality in including all packets s1 to sV [n] .
Hence we believe that there is no loss of generality in
restricting our attention to full-rank codes.
Theorem 1. The optimal throughput-delay trade-off among
full-rank codes is (τ, λ) = (r, D(r||p)) for all 0 ≤ r ≤ p. It
is achieved by the coding scheme with V [n] = ⌈rn⌉ for all n.
The term D(r||p) is the binary information divergence
function which is defined for 0 < p, r < 1 as
r
1−r
D(r||p) = r log + (1 − r) log
.
(3)
p
1−p
Note that as r → 0, D(r||p) converges to − log(1 − p), which
is the optimal λ, as given by Lemma 1.
To prove Theorem 1, we first show that the scheme with
transmit index V [n] = ⌈rn⌉ in time slot n achieves the tradeoff (τ, λ) = (r, D(r||p). Then we prove the converse by
showing that no other full-rank scheme gives a better trade-off.
Proof of Achievability: Consider the scheme with transmit index V [n] = ⌈rn⌉, where r represents the rate of adding
new packets to the transmitted stream. The rate of adding
packets is below the capacity of the erasure channel. Thus
it is easy to see that the throughput τ = r. Let E[n] be the
number of combinations, or equations received until time n. It
follows the binomial distribution with parameter p. All packets
s1 · · · sV [n] are decoded when E[n] ≥ V [n]. Define the event
Gn = {E[n] < V [n] for all 1 ≤ j ≤ n}, that there is no
packet decoding until slot n. The tail distribution of time T
between successive in-order decoding instants is,
⌈nr⌉−1
Pr(T > n) =
X
Pr(E[n] = k) Pr(Gn |E[n] = k),
k=0
⌈nr⌉−1 n k
=
p (1 − p)n−k Pr(Gn |E[n] = k),
k
k=0
⌈nr⌉−1 n k
. X
=
p (1 − p)n−k ,
(4)
k
k=0
n
.
=
p⌈nr⌉−1 (1 − p)n−⌈nr⌉+1 ,
(5)
⌈nr⌉ − 1
.
= e−nD(r||p) ,
(6)
X
where in (4), we remove the Pr(Gn |E[n] = k) when we
.
take the asymptotic equality = because, by the Generalized
Ballot theorem from [11], we can show that Pr(Gn |E[n] = k)
is ω(1/n). Hence it is sub-exponential and does not affect
the exponent of Pr(T > n). In (5), we only retain the
k = ⌈nr⌉ − 1 term from the summation because for r ≤ p,
that term asymptotically dominates other terms. Finally, we
use the Stirlings approximation of the binomial coefficient
n
nH(k/n)
to obtain (6).
k ≈ e
Hence we have proved that the scheme with V [n] = ⌈rn⌉
achieves the throughput-delay trade-off (τ, λ) = (r, D(r||p).
Proof of Converse: First let us show that the transmit
index V [n] of the optimal full-rank scheme should be nondecreasing in n. Given a scheme which does not satisfy the
non-decreasing property, we can permute the order of transmitting the coded packets such that V [n] is non-decreasing
in n. Changing the order of the transmitted packets will not
affect the throughput τ . And it can in fact improve the in-order
decoding exponent λ because decoding can occur sooner when
the initial coded packets include fewer source packets.
In the proposed scheme with V [n] = ⌈rn⌉, we add new
packets to the transmitted stream at a constant rate r. But in
general a full-rank scheme can vary the rate of adding packets.
Suppose it uses ratePri for ni slots for all 1 ≤ i ≤ L, such that
P
L
L
i=1 ni ri = nr. Then, the tail distribution
i=0 ni = n and
of time T between successive in-order decoding instants is,
P
⌈ L
i=1 ni ri ⌉−1
X
Pr(T > n) =
Pr(E[n] = k) Pr(Gn |E[n] = k),
k=0
(7)
⌈nr⌉−1 . X
=
.
=e
n k
p (1 − p)n−k ,
k
k=0
−nD(r||p)
.
(8)
(9)
Varying the rate of adding packets affects the term
Pr(Gn |E[n] = k) in (7), but it is still ω(1/n) and we can
eliminate it when we take the asymptotic equality in (8). As
a result, the in-order delay exponent is same as that if we
had a constant rate r of adding new packets to the transmitted
stream. Hence we have proved that no other full-rank scheme
can achieve a better (τ, λ) trade-off than V [n] = ⌈nr⌉ for all
n.
Fig. 1 shows the (τ, λ) trade-off for the immediate feedback
and no feedback cases, with success probability p = 0.6. The
optimal trade-off with any feedback delay d lies in between
these two extreme cases.
V. G ENERAL B LOCK - WISE F EEDBACK
In Section III and Section IV we considered the extreme
cases of immediate feedback (d = 1) and no feedback (d =
∞) respectively. We now analyze the (τ, λ) trade-off with
general block-wise feedback delay of d slots. We restrict our
attention to a class of coding schemes called time-invariant
schemes, which are defined as follows.
4
Block 1
In−order Decoding Exponent λ
1
Immediate
Feedback
(p, −log(1−p))
0.8
1
Block 3
Block 2
1
1
1
2
2
2
3
3
3
2
2
2
2
4
4
4
4
5
6
6
6
2
2
4
4
5
5
2
2
0.6
No Feedback
(r, D(r || p))
0.4
0.2
0
0
0.1
0.2
0.3
0.4
Throughput τ
0.5
0.6
Fig. 1. The trade-off between in-order decoding exponent λ and throughput
τ with success probability p = 0.6 for the immediate feedback (d = 1) and
no feedback (d = ∞) cases.
Definition 4 (Time-invariant schemes). A time-invariant
scheme is represented by a vector x = [x1 , · · · xd ]Pwhere xi ,
for 1 ≤ i ≤ d, are non-negative integers such that i xi = d.
In each block we transmit xi linear combinations of the i
lowest-index unseen packets in the stream.
The above class of schemes is referred to as time-invariant
because the vector x is fixed across all blocks. Observe that
as d → ∞, the class of time-invariant schemes are equivalent
to full-rank codes defined in Definition 3.
Conjecture 2. For any coding scheme, there exists a corresponding time-sharing policy between time-invariant schemes
that gives the same or strictly better (τ, λ) trade-off.
We believe this conjecture is true because, it can be shown
that any full-rank code can be expressed a time-sharing timeinvariant scheme. By Conjecture 1 it follows that there is no
loss of generality in focusing on time-invariant schemes.
There is also no loss of generality in restricting the length
of the vector x to d. This is because we are still transmitting d
independent coded packets. And adding fewer source packets
to the coded combinations, can only increase the exponent λ.
A. Analyzing the (τ, λ) of time-invariant schemes
Given a vector x, define pd , as the probability of decoding
the first unseen packet during the block, and Sd as the number
of innovative coded packets that are received during that block.
We can express τx and λx in terms of pd and Sd as,
E[Sd ] 1
(τx , λx ) =
, − log(1 − pd ) ,
(10)
d
d
where we get throughput τx by normalizing the E[Sd ] by the
number of slots in the slots. We can show that the probability
Pr(T > kd) of no in-order packet being decoded in k blocks
is equal (1 − pd )k . Substituting this in (1) we get λx .
Example 1. Consider the time-invariant scheme x =
[1, 0, 3, 0] where block size d = 4. That is, we transmit 1
combination of the first unseen packet, and 3 combinations of
Fig. 2. Illustration of the time-invariant scheme x = [1, 0, 3, 0] with block
size d = 4. Each bubble represents a coded combination, and the numbers
inside it are the indices of the source packets included in that combination.
The check and cross marks denote successful and erased slots respectively.
The packets that are “seen” in each block are not included in the coded packets
in future blocks.
the first 3 unseen packets. Fig. 2 illustrates this scheme for
one channel realization. The probability pd and E[Sd ] are,
3 3
pd = p + (1 − p)
p (1 − p)0 = p + (1 − p)p3 , (11)
3
3
X
4 i
i·
p (1 − p)4−i + 3p4 = 4p − p4 ,
(12)
E[Sd ] =
i
i=1
where in (12), we get i innovative packets if there are i
successful slots for 1 ≤ i ≤ 3. But if all 4 slots are successful
we get only 3 innovative packets. We can substitute (11) and
(12) in (10) to get the (τ, λ) trade-off.
Remark 1. Time-invariant schemes with different x can be
equivalent in terms of the (τ, λ). In general, given x1 ≥ 1, if
any xi = 0, and xi+1 = x ≥ 1, then the scheme is equivalent
to setting xi = 1 and xi+1 = x−1, keeping all other elements
of x the same. For example, x = [1, 1, 2, 0] gives the same
(τ, λ) as x = [1, 0, 3, 0].
B. Cost of Achieving Optimal τ or λ
In Section III we saw that for the immediate feedback case,
we can achieve (τ, λ) = (p, − log(1 − p)). However, when the
feedback is delayed we can achieve optimal τ (or λ) only at
the cost of sacrificing the optimality of the other metric. We
now find the best achievable τ (or λ) with optimal λ (or τ ).
Lemma 2 (Cost of Optimal Exponent λ). For a feedback delay
of d slots, the best achievable throughput is τ = (1 − (1 −
p)d )/d, when the in-order decoding exponent λ = − log(1 −
p).
Proof: If we want to achieve λ = − log(1−p), we require
pd in (10) to be equal to 1−(1−p)d. The only scheme that can
achieve this is x = [d, 0, · · · , 0], where we transmit d copies
of the first unseen packet. The number of innovative packets
Sd received in every block is 1 with probability 1 − (1 − p)d ,
and zero otherwise. Hence, the best achievable throughput is
τ = (1 − (1 − p)d )/d with optimal λ = − log(1 − p).
This result gives us insight on how much bandwidth (which
is proportional to 1/τ ) is needed for a highly delay-sensitive
application which needs λ to be as large as possible.
5
Optimal λ = −log(1−p)
0.6
Throughput τ
0.5
0.4
0.3
0.2
0.1
0
1
In−order Decoding Exponent λ
0.7
10
20 30 40
Block size d
50
Optimal τ = p
block using scheme x(1) and x(2) respectively. Suppose in an
interval with k blocks, we use scheme x(1) for h blocks, and
scheme x(2) in the remaining blocks, we have
0.8
Pr(T > kd) = (1 − pd1 )h (1 − pd2 )k−h .
0.6
(13)
Using this we can evaluate λ as,
0.4
k−h
h
+ λx(2) lim
k→∞
k
k
+ (1 − µ)λx(2)
λ = λx(1) lim
(14)
= µλx(1)
(15)
k→∞
0.2
0
10
20 30 40
Block size d
50
Fig. 3. Plot of the best achievable τ (or λ) versus d, while maintaining the
optimal value of the other metric λ (or τ ), for channel success probability
p = 0.6.
Lemma 3 (Cost of Optimal Throughput τ ). For a feedback
delay of d slots, the best achievable in-order decoding exponent is λ = − log(1 − p)/d, when the throughput τ = p.
Proof: If we want to achieve τ = p, we need to guarantee
an innovation packet in every successful slot. The only time
invariant scheme that achieve this is x = [1, 1, · · · 1], and the
vectors x that are equivalent to it as given by Remark 1.
With x = [1, 1, · · · 1], the probability of decoding the first
unseen packet is pd = p. Substituting this in (10) we get
λ = −log(1−p)/d, the best achievable λ when τ = p.
Fig. 3 shows the best achievable τ and λ versus d, when
the other metric is at its optimal value. The plots in Fig. 3
correspond to moving leftwards and downwards respectively
from the optimal trade-off (p, − log(1 − p)) in Fig. 1.
C. Finding Optimal (τ, λ) Trade-off
For any given throughput τ , our aim is to find the transmission scheme that achieves the maximum λ. We first prove
that any convex combination of achievable points (τ, λ) can
be achieved.
Theorem 2 (Convex Combinations of Time-invariant
Schemes). Given time-invariant schemes x(i) for 1 ≤ i ≤ B,
we can achieve the throughput-delay trade-off given by any
convex combination of the points (λx(i) , τx(i) ) by time-sharing
between the schemes.
Proof: Here we prove the result for B = 2, that is
time-sharing between two schemes. It can be extended to
general B using induction. Given two time-invariant schemes
x(1) and x(2) which achieve the throughput-delay trade-offs
(λx(1) , τx(1) ) and (λx(2) , τx(2) ) respectively, consider a timesharing strategy where, in each block we use the scheme
x(1) with probability µ and scheme x(2) otherwise. Then,
it is easy to see that the throughput on the new scheme is
τ = µτx(1) + (1 − µ)τx(2) .
Now we prove the in-order decoding exponent λ is also
a convex combinations of λx(1) and λx(2) . Let pd1 and pd2
be the probabilities of decoding the first unseen packet in a
where we get (14) using (10). As k → ∞, by the weak law
of large numbers, the fraction h/k converges to µ. Hence, we
have shown that we can interpolate between the (τ, λ) tradeoff of two policies by time-sharing between them.
The main implication of Theorem 2 is that, to find the
optimal (τ, λ) trade-off, we only have to find the points
(τx , λx ) that lie on the convex envelope of the achievable
region spanned by all possible x. We determine this optimal
trade-off for d = 2, 3 in Lemma 4 and Lemma 5 below.
Lemma 4 (Optimal Trade-off for d = 2). The
optimal (τ, λ) trade-off is the line joining points
(1 − (1 − p)2 )/2, − log(1 − p) and (p, − log(1 − p)/2).
Proof: When d = 2 there are only two possible timeinvariant schemes x = [2, 0] and [1, 1] that give unique (τ, λ).
By Remark 1, all other valid vectors x are equivalent to one
of these schemes. From Lemma 2 and Lemma 3 we know that
the (τ, λ) for these schemes are ((1−(1−p)2 )/2, − log(1−p))
and (p, − log(1 − p)/2) respectively. By Theorem 2 we can
achieve all (τ, λ) on the line joining these two points by timesharing between the two policies.
Lemma 5 (Optimal Trade-off for d = 3). The optimal (τ, λ)
trade-off when d = 3 is the piecewise linear curve joining
points
1 − (1 − p)3
(τA , λA ) =
, − log(1 − p) ,
(16)
3
2p(2 − p) 2
(τB , λB ) =
, − log(1 − p) ,
(17)
3
3
log(1 − p)
.
(18)
(τC , λC ) = p, −
3
Proof: When d = 3 there are four time-invariant schemes
x(1) = [3, 0, 0], x(2) = [2, 1, 0], x(3) = [1, 2, 0] and x(4) =
[1, 1, 1] that give unique (τ, λ), as given by Definition 4 and
Remark 1. From Lemma 2 and Lemma 3 we know that
(τx(1) , λx(1) ) = (τA , λA ) and (τx(4) , λx(4) ) = (τC , λC ).
For the other two schemes, we first evaluate pd and E[Sd ]
and substitute them in (10) to get, (τx(2) , λx(2) ) = (τB , λB )
and
(τx(3) , λx(3) ) = (3p − p3 )/3, −(log(1 − p)2 (1 + p))/3 .
We can show that x(2) gives a better trade-off than x(3)
by showing that for all p, the slopes of the lines joining
6
1
1
(p, −log(1−p))
(p, −log(1−p))
0.8
In−order Decoding Exponent λ
In−order Decoding Exponent λ
0.9
d =2
d =3
0.6
0.4
(r, D(r || p))
0.2
0.8
0.1
0.2
0.3
0.4
Throughput τ
0.5
0.5
0.4
0.3
0.2
0
0
0.6
Fig. 4. The throughput-delay trade-off with p = 0.6 for d = 2, 3 which
can be shown to be optimal over all convex combinations of time-invariant
schemes. The point just below the piece-wise linear curve for d = 3,
corresponding to the sub-optimal scheme x = [1, 2, 0].
(τx(i) , λx(i) ) for i = 1, · · · 4 satisfy,
λx(1)
τx(1)
λx(2)
τx(2)
λ (1)
− λx(2)
≥ x
τx(1)
− τx(2)
λx(3)
− λx(4)
≤
τx(3)
− τx(4)
− λx(3)
− τx(3)
− λx(4)
.
− τx(4)
(19)
(20)
The trade-off for d = 2 and d = 3 with p = 0.6 is shown in
Fig. 4. The point below the piece-wise linear curve for d = 3,
corresponding to the sub-optimal scheme x(3) = [1, 2, 0]. We
observe that the optimal trade-off becomes significantly worse
are d increases. From this we can imply that frequent feedback
to the source is important in delay-sensitive applications to
ensure fast in-order decoding of packets.
For general d, it is hard to search for the (τx , λx ) that lie
on the optimal trade-off. We suggest a set of time-invariant
schemes which are easy to analyze and they give a good (τ, λ)
trade-off.
Definition 5 (Suggested Schemes for General d). For general
d we suggest schemes with x1 = a and xd−a+1 = d − a, for
a = 1, · · · d. They give the throughput-delay trade-off
1 − (1 − p)a + (d − a)p a
, − log(1 − p) . (21)
(τ, λ) =
d
d
Fig. 5 shows the trade-off given by (21) for different values
of d. Observe that for d = 2 and d = 3 the suggested schemes
coincide with the optimal trade-off we derived in Lemma 4 and
Lemma 5 and shown in Fig. 4. As d → ∞, and a = αd, the
trade-off converges to ((1−α)p, −α log(1−p)) for 0 ≤ α ≤ 1,
which is the line joining (0, − log(1−p)) and (p, 0). Numerical
results suggest that for small d this class of schemes gives the
best trade-off among all possible time-invariant schemes x,
and close to optimal in general.
VI. C ONCLUDING R EMARKS
In this paper we analyze how block-wise feedback affects
the trade-off between throughput τ and in-order decoding
d =3
0.6
0.1
0
0
d =2
0.7
d=1
d=2
d=3
d=10
d=50
d=100
d=∞
0.1
(r, D(r || p))
0.2
0.3
0.4
Throughput τ
0.5
0.6
Fig. 5. The throughput-delay trade-off of the suggested coding schemes in
Definition 5 with p = 0.6 and different values of feedback delay d. Numerical
results suggest that this trade-off is optimal all convex combinations of timeinvariant schemes for small d.
exponent λ, which measures the burstiness in-order packet
decoding in streaming communication. When there is immediate feedback, we can simultaneously achieve the optimal τ
and λ. But as the block size increases, and the frequency of
feedback reduces, we have to compromise on at least one of
these metrics. Our analysis gives us the insight that frequent
feedback is crucial to ensure in-order packet delivery in delaysensitive applications.
Given that feedback comes in blocks of d slots, we present
a spectrum of coding schemes that span different points on the
(τ, λ) trade-off as shown in Fig. 5. Depending upon the delaysensitivity and bandwidth limitations of the applications, these
codes provide the flexibility to choose a suitable operating
point on trade-off. The proposed codes can be shown to be
optimal over the broad class of full-rank codes for small
feedback delay d, and when there is no feedback.
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